The Balanced Connected k-Partition Problem: Polyhedra and Algorithms

The balanced connected k-partition (BCPk) problem consists in partitioning a connected graph into connected subgraphs with similar weights. This problem arises in multiple practical applications, such as police patrolling, image processing, data base and operating systems. In this work, we address the BCPk using mathematical programming. We propose a compact formulation based on flows and a formulation based on separators. We introduce classes of valid inequalities and design polynomial-time separation routines. Moreover, to the best of our knowledge, we present the first polyhedral study for BCPk in the literature. Finally, we report on computational experiments showing that the proposed algorithms significantly outperform the state of the art for BCPk.

Robust Mixed 0-1 Programming and Submodularity

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.